I still feel like quests should accept any tier of dungeon (or Heroic at the worst) for convenience's sake, but having done my first Mythic (Eye of Azshara) just now, I am not really super worried about being able to complete any Mythic-required quests anymore. If you have no social network at all on WoW it probably sucks trying to do them, but my guild is so huge it's not that hard to round up four other people that would like to do a particular dungeon. Especially if there's a WQ on it that day. And they're certainly a stiffer challenge than heroic, enough that you probably wouldn't want to just charge in without any planning the way Dungeon Finder groups tend to. But we dropped 3 of the 5 bosses pretty much uneventfully and while our DPS went down on the giant boss our tank and healer managed to close the gap and finish it. It was only the giant snake that wiped us, and only once. That's apparently supposed to be one of the hardest Mythics, so I'm sanguine going forward.
What I really really don't want is anything that makes me do regular raids (aside from achievements, which I can always do in an expac or two). It's vastly harder to get raid groups together, especially since they have weekly lockouts, and while I haven't done PUG raiding since Wrath, progress was literally impossible with a PUG back then, and that was in ICC, which had a huge 30% buff going in. I don't expect it's gotten noticeably better since then. LFR may be baby mode, but I can guarantee I can get in and get it cleared. And that's what matters to me.